SMS scnews item created by Anthony Henderson at Fri 17 Nov 2006 1523
Type: Seminar
Distribution: World
Expiry: 24 Nov 2006
Calendar1: 24 Nov 2006 1205-1255
CalLoc1: Carslaw 373
CalTitle1: Algebra Seminar: Formanek -- Braid group representations of low degree

Algebra Seminar

Braid group representations of low degree

Edward Formanek

24th November, 12:05-12:55pm, Carslaw 373


Let G be a finitely generated group. For a fixed integer n, the equivalence classes of irreducible representations G -> GL(n,C) form an algebraic variety. There are only a few groups for which this variety is well understood. These include finite groups, abelian groups, abelian-by-finite groups and certain arithmetic groups, such as SL(n,Z) (n at least 3). Although classifying the irreducible representations for a general group is probably hopeless, the braid group Bn seems more tractable because its presentation is short and simple.

The combined work of E. Formanek, W. Lee, I. Sysoeva and M. Vazirani has classified the irreducible complex representations of Bn of degree <= n. Other than some exceptional representations when n <= 8, all such representations are either one-dimensional or the tensor product of a one-dimensional representation with a specialization of either the Burau representation or the standard representation. (The Burau and standard representations are representations Bn -> GL(r,C[t,t-1]), where r = n-2, n-1, or n, and t is an indeterminate. A specialization is a representation Bn -> GL(r,C) obtained by setting t equal to a nonzero complex number.)

After the seminar we will take the speaker to lunch.

See the Algebra Seminar web page for information about other seminars in the series.

Anthony Henderson

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